A note on connectivity in directed graphs

TL;DR

This paper determines the maximum number of edges in irredundant directed graphs on n vertices, characterizes the extremal cases, and resolves a question posed by Crane and Russell.

Contribution

It provides a complete characterization of the maximum edge count in irredundant directed graphs for all n, answering an open question in the field.

Findings

Maximum edges in irredundant directed graphs are characterized for all n.

The extremal graphs achieving maximum edges are fully described.

The work resolves a previously open problem by Crane and Russell.

Abstract

We say a directed graph $G$ on $n$ vertices is irredundant if the removal of any edge reduces the number of ordered pairs of distinct vertices $(u,v)$ such that there exists a directed path from $u$ to $v$. We determine the maximum possible number of edges such a graph can have, for every $n \in \mathbb{N}$. We also characterize the cases of equality. This resolves, in a strong form, a question of Crane and Russell.